Erdogan is an Islamist.
He’d like to be the new Caliph.
“There is no radical Islam, there is only Islam.” Turkish Prime Minister Erdogan
The question....”Turkey could cut off Islamic State’s supply lines. So why doesn’t it?”....
Because Turkey’s Erdogan has been buying the oil from ISIS. Simple...thus funding ISIS unquestionably.
Erdogan most likely does not desire to be Caliph. He might well hanker to be the Sultan. The Caliph only ruled in the first century. From thence he was the authority of religion and what he said went in that area. The Sultan ruled the geography and the society and the presence of the Caliph protected the sultan from attack a little bit. There was always a bit of implication that the Caliph was the hostage of the Sultan and it was not unknown for a sultan to depose a Caliph the Islamic way i.e. by strangling him or, in Mameluke Cairo, by halving him..